Is logical inconsistency (or fallibility) an inherent advantage of the human mind over computers? Does computing need to be logical? Minsky, in a quote reproduced in the paper, thought that, “There’s no reason to assume ... that either human minds or computing machines need to be perfectly and flawlessly logical.”

With the advent of heuristic computing, machine learning, statistical computing, and cognitive architectures like Watson, these questions are now more relevant than ever, but seldom discussed with academic rigor. Paradoxically, it was in the 1930s when the boundaries of computing and the nature of intelligent behavior were considered from a mathematical (logical) point of view.

This paper is part of this deep mathematical and logical stream of research that constitutes an important building block of a systematic analysis of the algorithmic capabilities of both man and machine. Its contribution is to remove some of the more controversial assumptions behind one of the arguments derived from Gödel’s second incompleteness theorem and formulated by Gödel himself in a 1951 speech. The argument is based on the conjecture that human mathematical reasoning is perfectly consistent. If that’s true, either there is no computer that can simulate human reasoning or, what is equally troubling, there is at least one statement related to arithmetic that human reasoning cannot master. In a word, the nature of mathematics is beyond the reach of an algorithmic computational entity, whether human or machine.

As deep as these statements are, the author forgoes philosophical speculation and focuses on a logical proof that eliminates the assumption that human reasoning needs to be consistent at all times. His arguments are directed to those already familiar with Peano arithmetic, Zermelo-Frankel set theory, and Gödel’s arguments in the proof of his first incompleteness theorem, and refer to previous works by the author. Still, the text is easy enough to follow with an unsophisticated knowledge of modern mathematical logic.

In summary, those interested in understanding the demonstrable boundaries of algorithmic reasoning will benefit from reading this work, which, as any rigorous theoretical work, is above technological fads.